Self-mixing interferometry (SMI), also known as optical feedback interferometry (OFI), is a well characterized method capable of measuring displacement related phenomena. The SMI effect was first observed in the decade of 1960, although it was then disregarded and considered as a nuisance for the development of laser based communications. Later, in the early 1980's, the interest of characterizing the behaviour of laser diodes (LD) gave rise to some of the early studies that can be related to the SMI method. Of those studies, probably the most representative is the study performed by Lang and Kobayashi [1]. The study describes and tests a mathematical model for a single-mode LD subject to feedback and the appearance of a modulation in the LD optical output power (OOP) related to the amount of feedback level. Later work on the subject proposed other models such as the double Fabry-Perot cavity [2] with a single reflection in the second cavity to account for the modulation produced over the OOP.
In short, SMI can be defined as the modulation of a LD-OOP caused when part of the laser light emitted (be1) is backscattered (br1) from the target (T) and re-enters the LD cavity, as shown in FIG. 1, interfering with the standing wave in the cavity. The changes in the OOP are then monitored either by an internal monitor photo-diode (PD), typically located at the back-facet of the LD electronic package, or by directly measuring voltage changes over the LD junction. Each of the schemes can show advantages and disadvantages as described in [3].
Mathematically, the SMI effect can be described by the following three equations:Δφ=(φF−φ0)τ+C sin(φFτ+a tan α),  (1)PF=P0[1+mF(φ)],  (2)F(φ)=cos(φFτ),  (3)where Δφ is the change of the LD phase due to the feedback, φ0 the initial LD phase, φF the phase after feedback, C the feedback level factor, α the linewidth enhancement factor, PF the optical output power after feedback, P0 the original LD output power and m the modulation factor.
In practice, the C factor shown in Eq. (1) has one of the most critical roles on the SMI signal interpretation, since it “controls” the shape of the signal and its suitability to perform accurate measurements. As discussed on [4] and [5], the value of C influences the SMI signal shape as follows:                If C<0.1: The SMI signal behaves in a purely sinusoidal fashion, and typically has small amplitudes. This is named the very weak feedback regime.        If 0.1<C<1: The SMI signal increases its amplitude and as C approaches to 1 the SMI signal acquires a sawtooth-like shape. This is named the weak feedback regime.        If C>1: The SMI signal has a sawtooth-like shape due to Hopf-bifurcation processes. As C increases, hysteresis appears and fringe loss situations can be expected. This is named the moderate feedback regime.        For large C values, the signal becomes chaotic and it cannot be used for measurement applications. This is named the coherence collapse regime.        
Thus, it is usually preferred to work within the weak (0.1<C<1) regime and close to the boundaries of moderate regime (C˜1) to avoid fringe-loss and keep a signal-to-noise ratio (SNR) higher than 10 dB, therefore simplifying the detection work and acquiring all the information that can be obtained from the SMI signal.
For the purpose of clarity, on further practical and theoretical descriptions presented on this specification it is considered that the LD used is a FP single-mode laser. Other types of lasers and LDs such as vertical cavity surface emitting lasers (VCSEL) and distributed feedback lasers (DFB) have already been tested and documented as suitable for SMI interferometry. Multi-mode lasers can also be applied for SMI, however the appearance of multiple harmonics in the sub-bands can enhance the difficulty of the signal processing. Two implementations of SMI using the method and system proposed by the present invention are discussed in a posterior section of the present specification since they may be applied using the differential self-mixing interferometry (DSMI) technique, namely the so-called “amplitude modulation embodiment” and the “current modulation embodiment.”
A differential self-mixing interferometry, or differential optical feedback interferometry (DOFI) method and system was proposed by the inventors in [6], as a solution to increase the SMI measurement resolution by comparing or operating two interferometric signals: the main measurement one generated by a first laser aiming to a target, and a reference SMI signal generated by another laser aiming the first laser or its mechanical holder while the first laser is moving.
Although the proposal made in [6] improved the resolution in comparison with the conventional SMI technique and enabled measuring very small features, it still has some drawbacks, such as the need of a second laser and its associated circuitry (such as a second amplification circuit and analogue-digital converter), which increases the cost of the final system and also the complexity of the operation thereof, as both lasers must perform as identically as possible to obtain the best results.
No considerations regarding the precision attainable in amplitude resolution are detailed in [6], where only some relationship between the phase and the measured displacement is established. Furthermore, the system in [6] requires a double-laser approach which, as stated above, implies a larger cost, more complex circuitry and processing, as far as the participation of two different lasers pushes up the requirements in order to ensure identical optical (wavelength, feedback level) and electronic (amplification circuitry, photodiode response) performances, and subsequently longer and more complex signal processing procedures. Such equivalence of the two lasers is impossible to attain in practical experimental configurations. Each difference in any parameter affecting the SMI signal reduces the accuracy of the technique as the reference and measurement signals will present small deviations from each other.
Finally, both references [10] and [8] disclose respective interferometric methods for measuring the optical path itself, i.e. to perform an absolute distance measurement, as any other known interferometric method (with the exception of the one described in [6]).
The methods presented on [10] and [8] consist on recovering the SMI signals from an electronically modulated laser and then estimating the absolute distance by relating the laser parameters. In order to perform that measurement, the target should be static during the signal acquisition.
No determination of the relative change in the optical path length between a laser and a target is disclosed at all in [10] nor in [8]. Obviously, such a relative change in the optical path length has nothing to do with a displacement measurement performed by a simple subtraction between the measurements of two absolute distances (not even such a simple subtraction measurement is disclosed in [10] nor in [8]).
Such a simple subtraction of absolute distances using SMI would not suffice to recover a displacement with a resolution comprised in a range between 0 and λ/2.
Furthermore, the caption of FIG. 9 in [10] mentions: “Mean and STD of measured distance as a function of target position in steps of 500 μm, Y axis corresponds to the absolute distance and X axis to the displacement”. Therefore, [10] defines displacement only as the difference between two static distances, not referring to the displacement of a target that is in motion during the measurement itself.
The resolution proposed by [10] and [8] is at least 5 orders of magnitude below said range between 0 and λ/2.
It is also important to keep in mind the differences between distance and displacement. Distance is a non-zero quantity that is equal to the total length of path from laser to target, therefore not containing any frequency or directional attribute. Displacement is a vectorial quantity that in magnitude only contains the differences in length, but also includes frequency and directional attributes.
It must also be pointed out that none of [10] and [8] disclose the use of a mechanical modulation, obviously because a mechanical modulation method has no capabilities of estimating the total absolute distance to the target, which is the goal of both [10] and [8].